Unlocking with residual-free bubbles Original Research Article

Residual-free bubbles are derived for the Timoshenko beam problem. Eliminating these bubbles the resulting formulation is form-identical in using the following tricks to the standard variational formulation: (i) one-point reduced integration on the shear energy term; (ii) replace its coefficient 1ϵ2 by 1(ϵ2 + (hK212)) in each element; (iii) modify consistently the right-hand side. This final formulation is ‘legally’ obtained in that the Galerkin method enriched with residual-free bubbles is developed using full integration throughout. Furthermore, this method is nodally exact by construction.